Zeroth-order parallel sampling

TL;DR

This paper introduces a novel zeroth-order parallel sampling method for MCMC that leverages multiple processors to achieve polynomial speed-up, addressing limitations of existing zeroth-order estimators.

Contribution

It proposes a new zeroth-order sampler using a random-slice approach, improving parallel efficiency over naive methods in MCMC.

Findings

Achieves polynomial speed-up with multiple processors

Outperforms naive zeroth-order estimators in parallel settings

Supported by theoretical analysis and numerical experiments

Abstract

Finding effective ways to exploit parallel computing to accelerate Markov chain Monte Carlo methods is an important problem in Bayesian computation and related disciplines. In this paper, we consider the zeroth-order setting where the unnormalized target distribution can be evaluated but its gradient is unavailable for theoretical, practical, or computational reasons. We also assume access to $m$ parallel processors to accelerate convergence. The proposed approach is inspired by modern zeroth-order optimization methods, which mimic gradient-based schemes by replacing the gradient with a zeroth-order stochastic gradient estimator. Our contribution is twofold. First, we show that a naive application of popular zeroth-order stochastic gradient estimators within Markov chain Monte Carlo methods leads to algorithms with poor dependence on $m$, both for unadjusted and Metropolis-adjusted schemes. We then propose a simple remedy to this problem, based on a random-slice perspective, as opposed to a stochastic gradient one, obtaining a new class of zeroth-order samplers that provably achieve a polynomial speed-up in $m$. Theoretical findings are supported by numerical studies.